It all depends
I had a somewhat unusual experience with a counter-offer. I was working for a contract firm on an option-to-hire contract with one of their clients. Both I and my employer knew that an offer would be forthcoming from the client, so my employer told me what sort of raise I would receive at the end of the contract, and it was a nice one that I thought was quite fair. I really did not want to become an employee of the client, so when they asked me what sort of salary I wanted, I gave them a figure that I really thought was much more than I was worth. They came up with an offer that met my request. I went back to my employer and told them of the offer and expressed that I would have a hard time turning it down, since it was about 20% more than what I would have received after my raise. I did not ask them to match, just possibly sweeten the promised raise, yet they matched the offer and continued to employ me for several more years. I guess they just had to increase my billing rate to cover my higher salary.
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